Count of pairs in Array such that bitwise AND of XOR of pair and X is 0
Last Updated :
28 Jul, 2021
Given an array arr[] consisting of N positive integers and a positive integer X, the task is to find the number of pairs (i, j) such that i < j and (arr[i]^arr[j] )&X is 0.
Examples:
Input: arr[] = {1, 3, 4, 2}, X = 2
Output: 2
Explanation:
Following are the possible pairs from the given array:
- (0, 2): The value of (arr[0]^arr[2])&X is (1^4)&2 = 0.
- (1, 3): The value of (arr[1]^arr[3])&X is (3^2)&2 = 0.
Therefore, the total count of pairs is 2.
Input: arr[] = {3, 2, 5, 4, 6, 7}, X = 6
Output: 3
Naive Approach: The simple approach to solve the given problem is to generate all possible pairs of the given array and count those pairs (i, j) that satisfy the given criteria i.e., i < j and (arr[i]^arr[j] )&X is 0.. After checking for all the pairs, print the total count obtained.
Below is the implementation of the above approach:
C++
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
int countOfPairs(int arr[], int N, int X)
{
// Stores the resultant count
// of pairs
int count = 0;
// Iterate over the range [0, N)
for (int i = 0; i < N - 1; i++) {
// Iterate over the range
for (int j = i + 1; j < N; j++) {
// Check for the given
// condition
if (((arr[i] ^ arr[j]) & X) == 0)
count++;
}
}
// Return the resultant count
return count;
}
// Driver Code
int main()
{
int arr[] = { 3, 2, 5, 4, 6, 7 };
int X = 6;
int N = sizeof(arr) / sizeof(arr[0]);
cout << countOfPairs(arr, N, X);
return 0;
}
// Java program for the above approach
class GFG
{
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
public static int countOfPairs(int arr[], int N, int X)
{
// Stores the resultant count
// of pairs
int count = 0;
// Iterate over the range [0, N)
for (int i = 0; i < N - 1; i++) {
// Iterate over the range
for (int j = i + 1; j < N; j++) {
// Check for the given
// condition
if (((arr[i] ^ arr[j]) & X) == 0)
count++;
}
}
// Return the resultant count
return count;
}
// Driver Code
public static void main(String args[])
{
int arr[] = { 3, 2, 5, 4, 6, 7 };
int X = 6;
int N = arr.length;
System.out.println(countOfPairs(arr, N, X));
}
}
// This code is contributed by gfgking.
# Python3 program for the above approach
# Function to find the number of pairs
# that satisfy the given criteria i.e.,
# i < j and (arr[i]^arr[j] )&X is 0
def countOfPairs(arr, N, X):
# Stores the resultant count
# of pairs
count = 0
# Iterate over the range [0, N)
for i in range(N - 1):
# Iterate over the range
for j in range(i + 1, N):
# Check for the given
# condition
if (((arr[i] ^ arr[j]) & X) == 0):
count += 1
# Return the resultant count
return count
# Driver Code
if __name__ == '__main__':
arr = [ 3, 2, 5, 4, 6, 7 ]
X = 6
N = len(arr)
print(countOfPairs(arr, N, X))
# This code is contributed by mohit kumar 29
// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG{
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
static int countOfPairs(int []arr, int N, int X)
{
// Stores the resultant count
// of pairs
int count = 0;
// Iterate over the range [0, N)
for(int i = 0; i < N - 1; i++)
{
// Iterate over the range
for(int j = i + 1; j < N; j++)
{
// Check for the given
// condition
if (((arr[i] ^ arr[j]) & X) == 0)
count++;
}
}
// Return the resultant count
return count;
}
// Driver Code
public static void Main()
{
int []arr = { 3, 2, 5, 4, 6, 7 };
int X = 6;
int N = arr.Length;
Console.Write(countOfPairs(arr, N, X));
}
}
// This code is contributed by SURENDRA_GANGWAR
<script>
// JavaScript program for the above approach
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
function countOfPairs(arr, N, X)
{
// Stores the resultant count
// of pairs
let count = 0;
// Iterate over the range [0, N)
for (let i = 0; i < N - 1; i++)
{
// Iterate over the range
for (let j = i + 1; j < N; j++)
{
// Check for the given
// condition
if (((arr[i] ^ arr[j]) & X) == 0)
count++;
}
}
// Return the resultant count
return count;
}
// Driver Code
let arr = [3, 2, 5, 4, 6, 7];
let X = 6;
let N = arr.length;
document.write(countOfPairs(arr, N, X));
// This code is contributed by Potta Lokesh
</script>
Time Complexity: O(N2)
Auxiliary Space: O(1)
Efficient Approach: The above approach can also be optimized by observing the given equation. So, to perform (A[i]^A[j]) & X == 0, then unset bits in the answer of (A[i]^A[j]) at the same position where the X has set bits in its binary representation is required.
For Example, If X = 6 => 110, so to make (answer & X) == 0 where answer = A[i]^A[j], the answer should be 001 or 000. So to get the 0 bit in the answer (in the same position as the set bit the X has), it is required to have the same bit in A[i] and A[j] at that position as the Bitwise XOR of the same bit (1^1 = 0 and 0^0 = 0) gives 0.
By closely looking at the relation between X and A[i] and A[j], it is found that X&A[i] == X&A[j]. Therefore, the idea is to find the frequency of the array elements having value arr[i]&X and any two numbers with the same value can be made as a pair. Follow the steps below to solve the problem:
- Initialize an unordered map, say M to store the count of numbers having a particular value arr[i]^X.
- Iterate over the range [0, N] using the variable i and increase the count of the value of arr[i]&X in the unordered map M.
- Initialize the variable count as 0 to store the resultant count of pairs.
- Iterate over the map M using the variable m and add the value of (m.second)*(m.second - 1)/2 to the variable count.
- After completing the above steps, print the value of count as the result.
Below is the implementation of the above approach:
C++
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
int countOfPairs(int arr[], int N, int X)
{
// Stores the resultant count
// of pairs
int count = 0;
// Initializing the map M
unordered_map<int, int> M;
// Populating the map
for (int i = 0; i < N; i++) {
M[(arr[i] & X)]++;
}
// Count number of pairs for every
// element in map using mathematical
// concept of combination
for (auto m : M) {
int p = m.second;
// As nC2 = n*(n-1)/2
count += p * (p - 1) / 2;
}
// Return the resultant count
return count;
}
// Driver Code
int main()
{
int arr[] = { 3, 2, 5, 4, 6, 7 };
int X = 6;
int N = sizeof(arr) / sizeof(arr[0]);
cout << countOfPairs(arr, N, X);
return 0;
}
// Java program for the above approach
import java.util.*;
class GFG{
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
static int countOfPairs(int[] arr, int N, int X)
{
// Stores the resultant count
// of pairs
int count = 0;
// Initializing the map M
HashMap<Integer,
Integer> M = new HashMap<Integer,
Integer>();
// Populating the map
for(int i = 0; i < N; i++)
{
if (M.containsKey(arr[i] & X))
M.put((arr[i] & X),
M.get(arr[i] & X) + 1);
else
M.put(arr[i] & X, 1);
}
// Count number of pairs for every
// element in map using mathematical
// concept of combination
for(Integer entry : M.keySet())
{
int p = M.get(entry);
// As nC2 = n*(n-1)/2
count += p * (p - 1) / 2;
}
// Return the resultant count
return count;
}
// Driver Code
public static void main(String[] args)
{
int[] arr = { 3, 2, 5, 4, 6, 7 };
int X = 6;
int N = arr.length;
System.out.print(countOfPairs(arr, N, X));
}
}
// This code is contributed by ukasp
# Python3 program for the above approach
# Function to find the number of pairs
# that satisfy the given criteria i.e.,
# i < j and (arr[i]^arr[j] )&X is 0
def countOfPairs(arr, N, X):
# Stores the resultant count
# of pairs
count = 0
# Initialize the dictionary M
M = dict()
# Populate the map
for i in range(0, N):
k = arr[i] & X
if k in M:
M[k] += 1
else:
M[k] = 1
# Count number of pairs for every
# element in map using mathematical
# concept of combination
for m in M.keys():
p = M[m]
# As nC2 = n*(n-1)/2
count += p * (p - 1) // 2
# Return the resultant count
return count
# Driver Code
if __name__ == '__main__':
arr = [ 3, 2, 5, 4, 6, 7 ]
X = 6
N = len(arr)
print(countOfPairs(arr, N, X))
# This code is contributed by MuskanKalra1
// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG{
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
static int countOfPairs(int []arr, int N, int X)
{
// Stores the resultant count
// of pairs
int count = 0;
// Initializing the map M
Dictionary<int,int> M = new Dictionary<int,int>();
// Populating the map
for (int i = 0; i < N; i++) {
if(M.ContainsKey(arr[i] & X))
M[(arr[i] & X)]++;
else
M.Add(arr[i] & X,1);
}
// Count number of pairs for every
// element in map using mathematical
// concept of combination
foreach(KeyValuePair<int, int> entry in M)
{
int p = entry.Value;
// As nC2 = n*(n-1)/2
count += p * (p - 1) / 2;
}
// Return the resultant count
return count;
}
// Driver Code
public static void Main()
{
int []arr = { 3, 2, 5, 4, 6, 7 };
int X = 6;
int N = arr.Length;
Console.Write(countOfPairs(arr, N, X));
}
}
// This code is contributed by ipg2016107.
<script>
// Javascript program for the above approach
// Function to find the number of pairs
// that satisfy the given criteria i.e.,
// i < j and (arr[i]^arr[j] )&X is 0
function countOfPairs(arr, N, X)
{
// Stores the resultant count
// of pairs
let count = 0;
// Initializing the map M
let M = new Map();
// Populating the map
for (let i = 0; i < N; i++) {
if (M.has(arr[i] & X)) {
M.set(arr[i] & X, M.get(arr[i] & X) + 1);
} else {
M.set(arr[i] & X, 1);
}
}
// Count number of pairs for every
// element in map using mathematical
// concept of combination
for (let m of M) {
let p = m[1];
// As nC2 = n*(n-1)/2
count += (p * (p - 1)) / 2;
}
// Return the resultant count
return count;
}
// Driver Code
let arr = [3, 2, 5, 4, 6, 7];
let X = 6;
let N = arr.length;
document.write(countOfPairs(arr, N, X));
// This code is contributed by gfgking.
</script>
Time Complexity: O(N)
Auxiliary Space: O(N)
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